Let \(n \in \mathbb{N}\) be the number of observations of a (log-)return
series \(\{r_t\}\), \(t=1,\dots,n\), and let \(\text{VaR}_t\) and
\(\text{ES}_t\) be the
estimated or forecasted VaR and ES (at some confidence level \(\alpha\)) at time
\(t\), respectively. Such series are included in an object of class
"fEGarch_risk". In the following, a risk measure at time \(t\) is
simply denoted by \(\text{RM}_t\) and can either mean
\(\text{VaR}_t\) or
\(\text{ES}_t\).
Based on a calculated VaR and / or expected shortfall (ES), capital needs
to be held back following regulatory rules. Commonly, among many models
used for forecasting risk measures that fulfill regulatory conditions,
loss functions are computed that also consider opportunity costs in to
assess, what model that fulfills regulatory rules minimizes such loss
functions. Let \(\Omega \geq 0\) be the penalty term.
For all loss functions we have
$$\text{LF}_i = \sum_{t=1}^{n} l_{t,i}, \hspace{3mm} i = 0,1,2,3,$$
as the loss function with
$$l_{t,i} = (\text{RM}_t - r_t)^2, \hspace{3mm} i = 0,1,2,3,$$
for \(r_t < \text{RM}_t\). They differ in how the case
\(r_t \geq \text{RM}_t\) is treated.
The regulatory loss function (rlf) uses \(l_{t,0} = 0\).
The firm's loss function (Sarma et al., 2003) (flf) considers
\(l_{t,1}=\Omega |\text{RM}_t|\).
The adjusted loss function (Abad et al., 2015) (alf) makes use
of \(l_{t,2} = \Omega |\text{RM}_t - r_t|\).
The corrected loss function (Feng, forthcoming) (clf) has
\(l_{t,3} = \Omega \text{min}\left(|\text{RM}_t - r_t|, |\text{RM}_t|\right)\).